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Temporal Point Processes have a long history of applications beginning with queueing theory (for telephone exchanges) in the 1910s and again in the 1980s (for communications networks); multi-electrode recordings from the 1950s, with a rapid expansion in the 1990s; genomics from the 2000s; high-frequency finance (e.g. modeling contagion) from the 2000s; and streaming data from the 2010s. A further explosion of applications is on the horizon due to the deployment of cheap sensors that emit signals only when signal changes exceed a threshold.
But the theory and modeling for (multivariate) point processes remains in its infancy. It lacks e.g. an infrastructure of tools comparable to that for multivariate time series. An increasingly popular approach to modeling self-exciting point processes (i.e. those with memory) is by means of Hawkes processes. But the existing theory assumes that data from the infinite past is available. For practical application the Hawkes process model must be truncated. We sketch the relevant system/stationarity theory and then discuss modeling of the Hawkes impulse response using Laguerre polynomials.
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